Jordan-Schwinger realizations of three-dimensional polynomial algebras

TL;DR

This paper introduces a method to construct three-dimensional polynomial algebras of arbitrary order using a generalized Jordan-Schwinger approach, expanding the algebraic toolkit for mathematical physics.

Contribution

It generalizes the Jordan-Schwinger realization to polynomial algebras of any order, enabling new algebraic constructions from commuting polynomial algebras.

Findings

Two commuting polynomial algebras can combine to form higher-order polynomial algebras.

The method extends the classical Jordan-Schwinger construction to polynomial algebras.

New algebraic structures can be systematically generated using this approach.

Abstract

A three-dimensional polynomial algebra of order $m$ is defined by the commutation relations $[P_0, P_\pm]$ $=$ $\pm P_\pm$, $[P_+, P_-]$ $=$ $\phi^{(m)}(P_0)$ where $\phi^{(m)}(P_0)$ is an $m$-th order polynomial in $P_0$ with the coefficients being constants or central elements of the algebra. It is shown that two given mutually commuting polynomial algebras of orders $l$ and $m$ can be combined to give two distinct $(l+m+1)$-th order polynomial algebras. This procedure follows from a generalization of the well known Jordan-Schwinger method of construction of $su(2)$ and $su(1,1)$ algebras from two mutually commuting boson algebras.